EDF6938
Blueprint for Exam 2
One-Way Between Subjects Analysis of Variance (ANOVA)
Assumptions: Normality (robust): if not normal, may lose power
Independence of Errors (not robust): if violated, use different test (within-subjects)
Homogeneity of Variance: if violated, use difference test (Welch’s V)
Source Table:
Source of Variation
Between Groups
Within Groups
Total
Hypothesis:
Sums of Squares
df
Mean Square
F
SSB
SSW
SST
k-1 (# of groups - 1)
N-k (sample size - # of groups)
N-1 (sample size - 1)
SSB / df (k-1)
SSW / df (N-k)
MSB / MSW
HO: M1 = M2 = M3
H1: Mi ≠ Mj (atleast one case is different – 2 tailed)
Critical Value: ** Use F Table in Text Book ** .05 sig level with both df values
Test Stat:
F (df, df) = 0.00, p = 0.00
Decision:
Reject/Fail to Reject (reject if p-value is less than sig level)
Conclusion:
“Results indicate there was / was not a significant difference between at least one of the groups.
According to the sample data… ”
Follow-Up:
Statistically Significant F: there is a difference, but where? Use contrast tests table (equal var)
HO: M1 - M2 = 0
HO: M1 – M3 = 0
Contrast: (1) M1 + (-1) M2 + (0) M3 = 0
Contrast: (1) M1 + (0) M2 + (-1) M3 = 0
Family-Wise Error Rate is the probability that one or more contrasts for a single factor will be
falsely declared sig, but the Per-Comparison Error Rate is the probability that a particular
contrast will be falsely declared sig. Controlling the Family-Wise Error Rate is one way to control
for inflated error.
Planned Contrasts are decided to test prior to examining results while Post Hoc Contrasts are
decided to test after examining results.
Bonferroni Adjustment: Adjusts per contrast Type I error rate so combined stays 0.05
One-Way Within Subjects Analysis of Variance (ANOVA)
Assumptions: Normality (robust): if not normal, may lose power
Sphericity: never assume all measures have same variance and that it holds – use correction.
Use Greenhouse-Geisser adjustment (adjusted df) to avoid inflated Type I error rate
Source Table:
Source of Variation
Treatment
Subjects
Error
Total
Hypothesis:
Sums of Squares
df
Mean Square
F
PROVIDED
a-1 ( # of treatments - 1)
s-1 (# per treatment level -1)
(a-1)(s - 1)
N-1 (total # of observs - 1)
SSa / df (a-1)
SSs / df (s-1)
MSTREAT/MSINTERACT
HO: M1 = M2 = M3 (# of treatments)
H1: Mj ≠ Mj prime
Critical Value: ** Use F Table in Text Book ** .05 sig level with both df values
Test Stat:
F (df, df) = 0.00, p = 0.00
Decision:
Reject/Fail to Reject (reject if p-value is less than sig level)
Conclusion:
“Results indicate there was / was not a significant difference between at least one of the groups.
According to the sample data… ”
Analysis of Covariance (ANCOVA)
Assumptions: Homogeneity of Variance
Normality
Independent of Error Terms
Equality of Slopes
Covariate is related to dependent variable
Covariate:
A continuous variable used to decrease variability in the error term in F ratio denominator
(F statistic if larger and more likely to be rejected) and it allows for infinite # of levels.
It is used to increase power and adjust for non-equivalent groups by making the groups more
homogeneous (adjust for mean differences).
Hypothesis:
HO: M1 = M2 = M3 (# of treatments)
H1: Mj ≠ Mj prime
Critical Value: ** Use F Table in Text Book ** .05 sig level with both df values
Test Stat:
F (df, df) = 0.00, p = 0.00 (df for task and error, not covariate)
Decision:
Reject/Fail to Reject (reject if p-value is less than sig level)
Conclusion:
“Results indicate there was / was not a significant difference between the treatments. According
to the sample data, the treatment is statistically significant and an association between … and …
holds for the population of … when controlling for … ”
Split-Plot
Assumptions: Between-Subjects
Normality
Homogeneity
Independence
Within-Subjects
Normality
Sphericity
Within-Subjects Independence
Hypothesis:
HO: no interaction (interaction)
HO: M1 = M2 (test)
HO: M1 = M2 (group)
H1: interaction
H1: M1 ≠ M2
H1: M1 ≠ M2
Critical Value: ** Use F Table in Text Book ** .05 sig level with both df values
Test Stat:
F (df, df) = 0.00, p = 0.00 (df for task and error, not covariate)
Decision:
Reject/Fail to Reject Each (reject if p-value is less than sig level)
Conclusion:
“There is / is not an interaction between the … and the … This means the mean difference in the
… was not contact across the … ”
Sphericity
Definition:
All measures have same variance and all correlations between any pair of measures are equal
When Used:
One-Way Within Subjects ANOVA and Split-Plot
Use Greenhouse-Geisser Adjustment (df, critical and p values adjusted)
Less likely to reject null when we should not